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Let an initial data metric $\overline{g}$ be, outside a ball $B_{R_0}$centered in the origin, the induced metric on $\Sigma_0$ of a Kerr spacetime(with a mass $M$ and angular momentum $J$ whose ratio, $J/M$, depends on thesize of $R_0$) plus small corrections which decay at spacelike infinity fasterthan $r^{-3}$; let, in the same region, a symmetric tensor $\overline{k}$ bethe second fundamental form of the Kerr spacetime plus small corrections whichdecay at spacelike infinity faster than $r^{-4}$, let $\overline{g}$ and$\overline{k}$ satisfy the constraint equations. Then, using the previousresults of \cite{Ch-Kl:book} and \cite{Kl-Ni:book}, the global existence of theexternal region of a global spacetime, outside the region of influence of$B_{R_0}$ follows. In this external region the various components of theRiemann tensor decay along the outgoing null directions in agreement with the"Peeling Theorem".

The peeling in the "very external region" of non linear perturbations of the Kerr spacetime